Demo 12

Kwang Hyun Kim

kkim@qcc.cuny.edu

Queensborough Community College

12/2/22

Right Triangle and Trigonometry

By Pythagorean theorem, \[x^2+y^2=r^2\]

\[\sin\theta=\frac{y}{r}\] \[\cos\theta=\frac{x}{r}\]

\[\tan\theta=\frac{\sin\theta}{\cos\theta}=\frac{y}{x}\] \[\csc\theta=\frac{1}{\sin\theta}=\frac{r}{y}\]

\[\sec\theta=\frac{1}{\cos\theta}=\frac{r}{x}\] \[\cos\theta=\frac{1}{\tan\theta}=\frac{x}{y}\]

Practice 1: No Decimal approximation!

import {make_positionC} from '@kkim/ma121'
pos=make_positionC(12,-5,'A')

Find the exact value of all six trigonometric functions of the angle \(A\).

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\(r\)
\(\sin,\csc\)
\(\cos,\sec\)
\(\tan,\cot\)

\[ r=\sqrt{\left(12\right)^2+\left(-5\right)^2}=\sqrt{169}=13\]

\[ \sin A=\frac{y}{r}={\color{red}-\frac{5}{13}},\csc A={\color{red}-\frac{13}{5}} \]

\[\cos A=\frac{x}{r}={\color{red}\frac{12}{13}},~ \sec A={\color{red}\frac{13}{12}}\]

\[\tan A=\frac{x}{y}={\color{red}-\frac{5}{12}},~ \cot A={\color{red}-\frac{12}{5}}\]

Practice 2: Exact values with a ref angle

Determine the exact value of \(\tan 120^\circ\) . (Decimal approximations will NOT be accepted.)

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\(120^\circ\) is in Quadrant II.
- \(\tan(120^\circ) < 0\)
- \(ref(120^\circ)=180^\circ-120^\circ= 60^\circ\) \[\tan 120^\circ=-\tan(60^\circ)=-\sqrt{3}\]